Sen Kuang

Lyapunov control methods of closed quantum systems

According to special geometric or physical meanings, the paper summarizes three Lyapunov functions in controlling closed quantum systems and their controller designing processes. Specially, for the average value-based method, the paper gives the generalized condition of the largest invariant set in the original reference and develops the construction method of the imaginary mechanical quantity; for the error-based method, this paper gives its strict mathematical proof train of thought on the asymptotic stability and the corresponding physical meaning. Also, we study the relations among the three Lyapunov functions and give a unified form of these Lyapunov functions. Finally, we compare the control effects of three Lyapunov methods by doing some simulation experiments.

Population Control of Equilibrium States of Quantum Systems via Lyapunov Method

Abstract This paper studies the population control problem associated with the equilibrium states of mixed-state quantum systems by using a Lyapunov function with degrees of freedom. The control laws are designed by ensuring the monotonicity of the Lyapunov function; main results on the largest invariant set in the sense of LaSalle are given; and the strict expression of any state in the largest invariant set is normally deduced in the framework of Bloch vectors. By analyzing the obtained largest invariant set and the Lyapunov function itself, this paper also discusses the determination problem of the degrees of freedom. Numerical simulation experiments on a three-level system show the validity of research results.

Rapid Lyapunov control of finite-dimensional quantum systems

Abstract Rapid state control of quantum systems is significant in reducing the influence of relaxation or decoherence caused by the environment and enhancing the capability in dealing with uncertainties in the model and control process. Bang–bang Lyapunov control can speed up the control process, but cannot guarantee convergence to a target state. This paper proposes two classes of new Lyapunov control methods that can achieve rapidly convergent control for quantum states. One class is switching Lyapunov control where the control law is designed by switching between bang–bang Lyapunov control and standard Lyapunov control. The other class is approximate bang–bang Lyapunov control where we propose two special control functions which are continuously differentiable and yet have a bang–bang type property. Related stability results are given and a construction method for the degrees of freedom in the Lyapunov function is presented to guarantee rapid convergence to a target eigenstate being isolated in the invariant set. Several numerical examples demonstrate that the proposed methods can achieve improved performance for rapid state control of quantum systems.

Lyapunov-Based Feedback Preparation of GHZ Entanglement of

The Greenberger–Horne–Zeilinger (GHZ) entangled states are a typical class of entangled states in multiparticle systems and play an important role in the applications of quantum communication and quantum computation. For a general quantum system of

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{N}

-Qubit Systems

qubits, degenerate measurement operators are often met, which cause the convergence obstacle in the state preparation or stabilization problem. This paper first generalizes the traditional quantum state continuous reduction theory to the case of a degenerate measurement operator and chooses a measurement operator for an arbitrarily given target GHZ entangled state, then presents a state stabilization control strategy based on the Lyapunov method and achieves the feedback preparation of the target GHZ state. In our stabilization strategy, we separate the target GHZ state and all the other GHZ states that often form the equilibrium points of the closed-loop system by dividing the state space into several different regions; and formally design a switching control law between the regions, which contains the control Hamiltonians to be constructed. By analyzing the stability of the closed-loop system in the different regions, we propose a systematic method for constructing the control Hamiltonians and solve the convergence problem caused by the degenerate measurement operator. The global stability of the whole closed-loop stochastic system is strictly proved. Also, we perform some simulation experiments on a three-qubit system and prepare a three-qubit GHZ entangled state. At the same time, the simulation results show the effectiveness of the switching control law and the construction method for the control Hamiltonians proposed in this paper.

An improved robust ADMM algorithm for quantum state tomography

In this paper, an improved adaptive weights alternating direction method of multipliers algorithm is developed to implement the optimization scheme for recovering the quantum state in nearly pure states. The proposed approach is superior to many existing methods because it exploits the low-rank property of density matrices, and it can deal with unexpected sparse outliers as well. The numerical experiments are provided to verify our statements by comparing the results to three different optimization algorithms, using both adaptive and fixed weights in the algorithm, in the cases of with and without external noise, respectively. The results indicate that the improved algorithm has better performances in both estimation accuracy and robustness to external noise. The further simulation results show that the successful recovery rate increases when more qubits are estimated, which in fact satisfies the compressive sensing theory and makes the proposed approach more promising.

Approximate bang-bang Lyapunov control for closed quantum systems

This paper proposes a new approximate bang-bang Lyapunov control that can achieve rapid state control for quantum systems. A construction method is presented to design the degrees of freedom in the Lyapunov function so that the control law can guarantee the convergence of the system to a target eigenstate being isolated in the invariant set. Simulation experiments on a three-level system demonstrate that the proposed method can achieve good performance for rapid state control of quantum systems.

Implicit Lyapunov control of multi-control Hamiltonian systems based on state distance

For closed quantum systems, if controlled systems are strongly regular and all other eigenstates are directly coupled to a target state, then such control systems can be asymptotically stabilized by means of the Lyapunov-based control. However, when the controlled systems are not strongly regular, or when there exists at least one eigenstate that is directly uncoupled to the target state, the situations will become complicated. This paper proposed a method based on an implicit Lyapunov function to overcome these two degenerate cases. Also, the proposed method is suitable for multi-control Hamiltonian systems. The convergence of closed-loop systems is analyzed by the LaSalle invariance principle. Finally, numerical simulation experiments on a 4-level system are done. The experiment results show the effectiveness of the implicit Lyapunov control method for degenerate cases and multi-control Hamiltonians.

Population control of quantum states based on invariant subsets under a diagonal Lyapunov function

This paper deduces and analyzes the invariant set in quantum Lyapunov control, explores the principles for constructing and adjusting diagonal elements of a diagonal Lyapunov function, and achieves the convergence to any goal state in some invariant subset of closed loop systems by using dynamical system theory and energy-level connectivity graph. Research results show that if a goal state is an eigenstate of the inner Hamiltonian, then it is very easy to achieve convergence to the goal state with a high probability; and if a goal state is a superposition state in some invariant subset, then it is possible to achieve satisfactory control when the diagonal elements are properly constructed.